COGNITIVE

l-38 (1983)

PSYCHOLOGY

15,

SchemaInduction

MARY

and AnalogicalTransfer

L. GICK AND KEITH J. HOLYOAK
Universityof Michigan

An analysis of the process of analogical thinking predicts that analogies will be
noticed on the basis of semantic retrieval cues and that the induction of a generalschema from concrete analogs will facilitate analogical transfer. These predictions
were tested in experiments in which subjects first read one or more stories illustrating problems and their solutions and then attempted to solve a disparate but
analogous transfer problem. The studies in Part I attempted to foster the abstraction of a problem schema from a single story analog by means of summarization
instructions, a verbal statement of the underlying principle, or a diagrammatic
representation of it. None of these devices achieved a notable degree of sucess. In
contrast, the experiments in Part II demonstrated that if two prior analogs were
given, subjects often derived a problem schema as an incidental product of describing the similarities of the analogs. The quality of the induced schema washighly predictive of subsequent transfer performance. Furthermore, the verbal
statements and diagrams that had failed to facilitate transfer from one analog
proved highly beneficial when paired with two. The function of examples inlearning was discussed in light of the present study.

Analogy pervades thought. When a John Donne proposes that “no man
is an island,” we feel an intuitive grasp of the interconnectedness ofhuman relations. When a William Harvey compares a biological organ to awater pump, a productive scientific model of blood circulation is created;in addition, the meaning of “pump” may take on a new, more abstract
form. When a student is told that the atom resembles a miniature solar
system, a complex new concept may take root in the learner’s mind. To
make the novel seem familiar by relating it to prior knowledge, to makeThis paper is largely based on a PhD dissertation completed by Gick (Note 1) under the
direction of Holyoak, together with additional collaborative experiments and analyses. The
work benefitted from the guidance of the members of the dissertation committee: John
Jonides, Manfred Kochen, David Krantz, and substitute member John Holland. Patricia
Cheng provided incisive criticisms of an early draft; a subsequent draft benefitted from thereviews of Dedre Gentner, Earl Hunt, and an anonymous referee. Terra Albert, Holly
Brewer, Tim Carroll, Teresa Frankovich, and Michael Smith ably assisted in testing subjects. Susan Petersen and Jean Schtokal assisted with both subject testing and scoring ofdata. Experiment 4 was reported at the 20th meeting of the Psychonomic Society, Phoenix,
November 1979. This research was supported by a University of Michigan Rackham Dissertation Grant to Gick, and by National Institute for Mental Health Grant S-ROlMH332878-03 and National Institute for Mental Health Research Scientist Development
Award 1-K02-MHOO342-02,both to Holyoak. Reprint requests may be sent to K. Holyoak,
University of Michigan, Human Performance Center, 330 Packard Road, Ann Arbor, MI
48104.
OOIO-0285183/010001-38$07.50/O
Copyright‘0 1983 by Academic Press, Inc.
All rights of reproduction
in any form reserved.

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GICK ANDHOLYOAKthe familiar seem strange by viewing it from a new perspective-these arefundamental aspects of human intelligence that depend on the ability toreason by analogy. This ability is used to construct new scientific models,
to design experiments, to solve new problems in terms of old ones, tomake predictions, to construct arguments, and to interpret literary
metaphors. (See Miller, 1979, for a recent defense of the classical viewthat metaphor should be analysed in terms of analogy.)
The essence of analogical thinking is the transfer of knowledge fromone situation to another by a process of mapping-finding a set of one-toone correspondences (often incomplete) between aspects of one body of
information and aspects of another. A central psychological issue concerns the mechanisms underlying analogical transfer. Reasoning by
analogy typically implies a comparison of two concepts (“analogs”) at thesame (usually quite concrete) level of abstraction (e.g., the heart and awater pump). However, a similar mapping process may be required tocompare a specific concept to a more general schema (e.g., the heart andthe abstract concept of “pump”). Furthermore, mapping may also be
involved in the induction of schemas from examples (e.g., learning theabstract sense of “pump” by comparing hearts and water pumps). Such aclose relationship between the processing of concrete analogs and generalschemas is supported both by experimental evidence (Schustack &Anderson, 1979) and computational analysis (Winston, 1980).’
The present study was designed to investigate the mechanisms thatgovern analogical transfer between semantically disparate problems. The
central empirical questions concern how analogies are noticed and thenapplied to generate solutions to novel problems. In addressing thesequestions the experiments explore the induction of a problem schemafrom concrete analogs and the role of the schema in fostering subsequenttransfer. To provide a context for the empirical work, we will first briefly
review our previous research on analogical problem solving and thenprovide a theoretical analysis of the nature of analogy and its relationshipto schema induction. In addition to generating predictions regarding thedeterminants of analogical transfer, this analysis may serve to clarify theconcept of “schema,” which has been widely applied in cognitive models,
but also widely criticized for its vagueness.
BACKGROUNDRESEARCH

Most psychological research on analogical reasoning has used fourterm “proportions” as stimuli (i.e., A:B::C:?; for a review see Stemberg,
1977). The adequacy of the proportion framework as a model of analogicalthinking has been questioned by Hesse (1966) and Weitzenfeld and Klein
1 The present distinction between a general schema and a specific analog is close to thatbetween the first and second senses of the term “model”
as it is used by Hesse (1966).

ANALOGICAL

TRANSFER

3(Note 2). A major limitation is that such stimuli obviate any need for the
subject to spontaneously notice the analogy, which is often a prerequisite
for successful transfer in realistic problem situations. A few studies haveinvestigated the use of analogies to solve problems. For example, Gentner
and Gentner (in press) have demonstrated that alternative analogies,
known by subjects prior to the experiment, produce systematically varying patterns of difficulty among types of electricity problems. Other
studies have indicated that potential analogies presented in the experimental context are often not used (Reed, Ernst, & Banerji, 1974) or evennoticed, especially when the problem analogs are from different semantic
domains (Duncker, 1926; Gick & Holyoak, 1980). We will discuss theGick and Holyoak study in more detail, since it led directly to the present
investigation. We had subjects attempt to solve Duncker’s (1945) “radiation problem,” which was the primary problem used in the present study:
Suppose you are a doctor faced with a patient who has a malignant
tumor in his stomach. It is impossible to operate on the patient, but unlessthe tumor is destroyed the patient will die. There is a kind of ray that canbe used to destroy the tumor. If the rays reach the tumor all at once at a
sufficiently high intensity, the tumor will be destroyed. Unfortunately, atthis intensity the healthy tissue that the rays pass through on the way tothe tumor will also be destroyed. At lower intensities the rays are harmless to healthy tissue, but they will not affect the tumor either. What typeof procedure might be used to destroy the tumor with the rays, and at thesame time avoid destroying the healthy tissue?
Prior to their attempt to solve the radiation problem, our subjects often
read a story about an analogous military problem and its solution (see“The General” in Appendix II). In this story a general wishes to capture afortress located in the center of a country. There are many roads radiating
outward from the fortress. All have been mined so that while small groupsof men can pass over the roads safely, any large force will detonate themines. A full-scale direct attack is therefore impossible. The general’ssolution is to divide his army into small groups, send each group to thehead of a different road, and have the groups converge simultaneously on
the fortress. As you may have already noticed, there is an analogous“convergence” solution to the radiation problem. The doctor could direct
multiple low-intensity rays toward the tumor simultaneously from different directions, so that the healthy tissue will be left unharmed, but theeffects of the low-intensity rays will summate and destroy the tumor.*
2 In our earlier paper (Gick and Holyoak, 1980) we referred to this as the “dispersion”
solution, emphasizing the initial division of the single large force into several small ones andtheir dispersal to multiple locations. However, in some forms of the problem a single large
force may not initially exist (see “Red Adair” in Appendix II), making our original name forthe solution quite misleading. Calling it the convergence solution emphasizes its central
property, the convergence of multiple forces on the target.

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GICK

AND

HOLYOAKA paradigm for investigating analogical problem solving must satisfy
two basic requirements. First, a relevant known analog must be available
to the subject. Provision of a story analog prior to the radiation problem
fulfilled this prerequisite. Second, the target problem must be sufficiently
novel and challenging that an analogy could potentially be useful. Subjects are unlikely to bother to apply an analogy if a solution to the target
problem is already known or can be easily achieved by means-ends
analysis. The radiation problem, due to its ill-defined nature, meets thesecond requirement. The problem admits of a variety of potential solutionplans, and we were able to exploit this feature to demonstrate the influence of an analogy. By varying the solution provided in the military story,
we were able to selectively facilitate the discovery of various alternative
potential solutions to the radiation problem. The influence of the analogy
was very pronounced. For example, about 75% of those subjects who
received the appropriate military analog produced the convergence solution to the radiation problem, as compared to less than 10% of control
subjects who did not receive the analogy.
The above results were obtained when subjects were given an explicit
hint to use the prior story to help solve the radiation problem. The hint
was quite nonspecific, as subjects were told nothing about how the story
might help; however, it obviated any need for subjects to spontaneously
notice the potential analogy. In further experiments we attempted to assess subjects’ ability to notice the analogy separately from their ability tosubsequently apply it to generate an analogous solution to the target
problem. The study of noticing raises some tricky methodological issues.
Ideally, one would like to provide an ecologically natural context for theprior analog; however, the range of such contexts is not readily delimited.
Subjects must be led to process the analog in such a way that the information will be potentially retrievable when they later encounter the target
problem. Task demands should neither make it obvious to subjects thatthey should use the analogy, nor preclude the possibility of their noticing
its relevance.
To approximate the above requirements we had subjects first memorize
the military story analog in the guise of a study of story recall and then
immediately go on to work on the radiation problem, without any hint to
use the prior story. Under these conditions only about 30% of our subjects
produced the convergence solution, as opposed to about 75% who produced it when a hint was given. Assuming that about 10% of subjects
would produce the solution in the absence of any analogy (a figure basedon several replications), these results indicate that only a third or less ofthe subjects who could potentially apply the analogy spontaneously
noticed it. We found this striking gap between noticing and application
quite surprising, although, as noted above, comparable outcomes have

ANALOGICAL

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5

been obtained in other transfer paradigms (Kohler, 1940; Reed et al.,1974). Our procedure did not involve any deception; subjects were told at
the outset that the experiment would have two parts, story recall andthen problem solving. The delay between the two tasks was minimal. One
might well have supposed that the demand characteristics of being in a
psychology experiment would have led virtually all subjects to consider
how the first part might be related to the second. Indeed, a participant in
one of our earlier experiments had complained to us that giving a hint to
use the story was an insult to our subjects’ intelligence!
It is thus no easy matter to spontaneously notice an analogy between
two superficially dissimilar situations, even in our highly simplified experimental paradigm. The present study investigates factors that influence
the likelihood that subjects will notice and apply analogies. To derive
specific predictions we first need to analyze the structure of analogy andthe relationship between specific analogs and more abstract schemas.
ANALOGICAL
Structure

THINKING

AND SCHEMASof Analogy

It is important to recognize that the structure of analogy is dictated by
its function. In analogical problem solving, one problem and its solution
are already known. The analogist notes correspondences between the
known problem and a new unsolved one, and on that basis derives ananalogous potential solution. More generally, the function of an analogy is
to derive a new solution, hypothesis, or prediction; this is done by finding
an initial partial mapping between the two analogs and then extending the
mapping by retrieving or creating additional knowledge about the analog
that was initially less well understood.
Each analog is thus conceptually divided into two parts: that which
provides the initial basis for mapping, and that which constitutes the“conclusion.” As Hesse (1966) has argued, analogy involves two distinct
types of relationships: the “horizontal” mapping relationship between
aspects of the two analogs, and the “vertical” relationship between thetwo parts of a single analog. At the most general level, the latter relationship is between relevant antecedent conditions and their correlated consequences.3 In many cases, including analogies between problems, the3 Metaphors typically produce an initial partial mapping that creates an expectation that
additional correspondences can he found, involving correlated aspects of the referent. For
example, suppose we assert that “Analogy is a window on the mind.” The most salient
attribute of a window is probably that it is something one sees through; accordingly, in an
appropriate context (such as a psychological report) we might interpret the remark to meanthat studying analogy is a way to gain understanding of (“see”) the mind. The initial mapping then constitutes an “invitation” to pursue the metaphor further. Thus, one might notethat a window is typically a relatively small opening in an opaque wall, suggesting that theGICK AND

6

HOLYOAKvertical relationships will correspond to causal relations within the person’s mental model of each situation (Winston, 1980). For example, certain aspects of an initial problem situation will be viewed as sufficientconditions for the attainment of a particular solution. The vertical andhorizontal relationships in an analogy are inextricably linked. For while itis not essential to map all aspects of the two analogs (indeed, this will
seldom be possible), those aspects which constitute the causal
antecedents of the known outcome obtained in the referent analog must
be mappable. If these causal elements cannot be mapped, the putative
analogy can be rejected as misleading.
Analogical

Mappingand Schema Induction

To understand the process of reasoning by analogy and the relationshipbetween an analog and a schema, it is necessary to examine the concept of“mapping” in some detail. While many important questions about the
mapping process must remain unanswered, several general points deserveemphasis. First, mappings may involve aspects of the analogs that havenever been explicitly presented to the analogist. Consequently, the various inference processes required for everyday understanding will often
play important roles in analogical thinking. Second, the mapping process
will often involve a search for “alternative views” of one of the analogs(Moore & Newell, 1973; Schon, 1979). For example, in order to use themilitary story to help solve the radiation problem, subjects must presumably view the story as a problem and its solution, rather than as, say, an
anecdote about a populist hero. In addition, mappings can be defined at
multiple levels of abstraction, which may correspond to “macrostructures” in the sense of Kintsch and van Dijk (1978) and van Dijk (1980).
The optimal level of representation will be that which maximizes thedegree of correspondence between causally relevant features of theanalogs. Finally, since mapped elements (relations and properties andtheir arguments) are typically similar but not identical, they must be decomposable into identities and differences (Hesse, 1966; Tversky, 1977).
To make our discussion more concrete, we present in Table 1 an informal description of the mapping relations between the military story andthe radiation problem. The correspondences are stated at a fairly abstractlevel of macrostructure (for a description at a more concrete level, see
Gick & Holyoak, 1980). We assume that each of the two analogs is an
instance of a very general “problem schema,” which is organized hierarmind is generally difftcult to study. Furthermore, a small window often allows only partial
view of what lies beyond, intimating that analogy may provide limited and selective insights,
and so on. The interpretation of metaphor thus seems to begin with detection of salient initialcorrespondences,followed by the construction of others (Jaynes, 1976, Chap. 2).

ANALOGICAL

Correspondences

TRANSFERTABLE 1among Two Convergence

7

Problems and Their Schema

Military problem
Initial state
Goal: Use army to capture forces.
Resources: Sufficiently large army.
Constraint: Unable to send entire army along one road.
Solution plan: Send small groups along multiple roads simultaneously.

Outcome: Fortress captured by army.
Radiation problem
Initial state
Goal: Use rays to destroy tumor.
Resources: Sufficiently powerful rays.
Constraint: Unable to administer high-intensity rays from one direction.
Solution plan: Administer low-intensity
rays from multiple directions simultaneously.
Outcome: Tumor destroyed by rays.
Convergence schema
Initial state
Goal: Use force to overcome a central target.
Resources: Sufficiently great force.
Constraint: Unable to apply full force along one path.
Solution plan: Apply weak forces along multiple paths simultaneously.
Outcome: Central target overcome by force.

chically into an initial state (goals, available resources, and constraints), a
solution plan, and an actual or anticipated outcome of realizing the plan.
The problem schema reflects the vertical organization of the analogs, as
we discussed earlier; the initial state includes relevant antecedent conditions, of which the solution plan and outcome are consequences. These
problem components thus have a natural procedural interpretation as“situation-action”
rules (Winston, 1980). The task of the analogist is toconstruct a partial mapping between the two initial states, which can beused to construct the analogous solution plan and expected outcome forthe radiation problem.
At the level of macrostructure assumed in Table 1 the two problems canbe completely mapped, i.e., a one-to-one correspondence can be foundbetween their components. For example, the goal of using the army tocapture the fortress maps onto the goal of using the rays to destroy thetumor. The basis of such a mapping relation is an abstract “core idea”
that both mapped propositions instantiate, which we will term a “mapped
identity.” (See Holyoak, in press, for a more detailed taxonomy of mapping relations.) In each problem the goal, for example, is to use a force toovercome a central target. At the bottom of Table 1 these mapped iden-

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GICK AND HOLYOAK

tities are abstracted and stated as a “convergence schema”; i.e., a representation of the type of problem for which convergence solutions arefeasible. Such a schema can be abstracted from the two analogs by“eliminative induction” (Mackie, 1974; see Winston, 1980, for a computational implementation). In essence, the process of schema induction
involves deleting the differences between the analogs while preserving
their commonalities.4
The schema can be viewed as an abstract category that the individual
analogs instantiate in different ways. For example, the goal of “overcoming a target” is instantiated as “capturing a fortress” in the militarystory, and as “destroying a tumor” in the radiation problem. Suchdomain-specific instantiations of the schema can be termed “structurepreserving differences” between the two analogs (Holyoak, in press). In
our example, the military analog can be viewed as a transformation of theconvergence schema into concepts appropriate to a military domain, andthe medical analog can be viewed as a transformation of the same schema
into concepts appropriate to a medical domain. Since the schema is implicitly embedded in each analog, and assuming the mapped identities arecausally sufficient to yield the outcome associated with the schema, it
follows that structure-preserving differences do not alter the causal relations in the schema. Such differences make the problems analogous rather
than identical.
The distinction between mapped identities and structure-preserving
differences helps to elucidate the intuitive notion that analogs differ intheir similarity to one another. The similarity of any pair of mapped concepts will increase with the extent of the meaning captured by a mapped
identity (Tversky, 1977). In general, if the mapped concepts are either
identical or instances of a close superordinate concept, the analogs will bevery similar (yielding, one might say, a “literal”
rather than a“metaphorical” analogy). Thus, a story about a doctor treating a brain
tumor with multiple lasers would obviously be more similar to the radiation problem than is our military problem. A “deep” analogy, the sortthat captures our admiration, is an analogy between disparate situations inwhich the essential causal relations are maintained.
Differences between analogs are not always structure preserving. A“structure-violating”
difference is one that does not conform to the general transformation relating the schema to its analog (Holyoak, in press).
Such differences make the analogy incomplete. While our convergence
4 In the case of metaphor, the formation of a schema from mapped identities, so that the
subject and referent can be viewed as instances of a more general concept, may provide abasis for the “interaction view” of metaphorical interpretation (Black, 1962).

ANALOGICAL

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9analogy is complete at the level of macrostructure represented in Table 1, it
is incomplete at more specific levels (see Gick & Holyoak, 1980). For
example, the role of the army in the military story usually corresponds to
the role of the rays in the medical problem. But whereas a direct attack on
the fortress would endanger the army, direct treatment of the tumor
would not endanger the rays. (Rather, it would harm healthy tissue). Thisviolation of role parallelism constitutes a structure-violating difference
between the two analogs. Normatively, a structure-violating difference
should lead to rejection of the analogy as inappropriate if such a difference
alters an element causally necessary for the solution to the known problem (which is not the case for the above example).
In general, increasing the level of representational abstraction will increase the completeness of an analogy, by deleting mismatching details(both structure-preserving and structure-violating differences). Note,however, that greater completeness need not be entirely a virtue. For
example, one might claim that any two problems are completely analogous at the level of the abstract problem schema corresponding to theheadings used in Table 1. But such abstract analogies will seldom trigger
development of a realizable solution procedure. In general, increasing thelevel of abstraction will at some point delete corresponding identities aswell as mismatches, and consequently diminish the similarity of theanalogy. A tendency to maximize the completeness of an analogy bymoving to a more abstract level of macrostructure may therefore often
compete with a tendency to maximize the extensiveness of the mapping
between causal relations by moving to a more detailed representational
level. As a result, the “optimal” level of representation for successful
analogical thinking may typically lie at an intermediate level of abstraction, and it may yield an analogy that is less than complete.
Schemas as Mediatorsof AnalogicalTransfer

The above analysis suggests how analogical transfer may take place.
The schema (mapped identities) affords the basis for analogical transfer.
However, we can distinguish two conceptually distinct ways in which theschema could be used in solving a problem with reference to information
obtained from prior analogs. First, the new problem may be mapped
directly with a prior analog to generate the analogous solution. While themapped identities will mediate the transfer process, the schema need notexist as a separate concept independent of the two analogs (although an
independent schema may be incidentally induced in the course of the
mapping process). We will refer to this case as “reasoning from ananalog.” Second, an independent schema may already have been induced
from one or more prior analogs and stored in memory. The person can

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GICK ANDHOLYOAK

therefore map the new analog directly with the schema in order to construct a solution. This case will be termed “reasoning from a schema.”
There are several reasons why our analysis predicts a processing advantage for reasoning for a schema rather than an analog. Consider first
how an analogy might be initially noticed. It is well established thathuman memory search is guided by semantic retrieval cues. We might
suppose, then, that any semantic aspect of the novel problem could potentially provide a link to a relevant analog. There will be many potential
cues to retrieve a very similar problem from the same semantic domain;for example, the radiation problem is likely to call to mind prior knowledge about related medical procedures. But by its very nature, an analog
from a disparate domain will lack such transparent resemblances. Consequently, the potential semantic links between two dissimilar analogs
will simply correspond to the basis of the analogy: the identities thatcomprise the implicit schema embedded within each analog.
It follows that if an appropriate schema has not been at least partiallyabstracted, it will be relatively difficult to retrieve a prior analog when
given the target problem, because it is the schema that affords potential
retrieval cues. Tversky’s (1977) analysis of similarity implies that an
analog will be more similar to its schema than to another analog, because
the schema contains all the aspects common to the two analogs (mapped
identities) and none of the differences between them. An independent
schema will therefore facilitate the retrieval and noticing of an analogy.
In addition, it should be easier to apply a schema than an analog. An
explicit problem schema will make salient those causal aspects of a situation that should trigger a particular plan of action. When two analogs are
drawn from disparate domains, the inference processes underlying the
mapping process will be difficult to execute; if the optimal mapping istherefore not found, the analogist may fail to generate the corresponding
solution to the target problem. In general, mapping an analog to a schema
will be simpler than mapping one analog with another, because in theformer case it will only be necessary to map identities, rather than both
identities and differences. We therefore predict that factors favoring
schema induction will facilitate generation of the intended solution even if
a “teacher” is available to call the person’s attention to the relevant
analogy.
OVERVIEW

OF EXPERIMENTS

Methodological Issues
The basic procedure used to separate spontaneous noticing of ananalogy from its application after a hint was adapted from Gick andHolyoak (1980) and was used in all of the present experiments. ThisANALOGICAL

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11

procedure has three steps: (1) subjects process a story analog in the guiseof a task such as story comprehension; (2) they then attempt to solve theradiation problem without any explicit hint to use the prior story; and (3)they are explicitly asked to propose a solution suggested by the story. The
percentage of subjects producing the analogous solution to the target
problem on their first pass provides a measure of spontaneous noticing
and application: the total percentage of subjects producing the solution oneither pass provides a measure of potential application given that theanalogy has been noticed. The difference between these two percentages
is a measure of the importance of having a “teacher” to point out therelevance of the prior analog.
This procedure is particularly sensitive to the effect of a hint, because
each subject attempts to solve the problem both before and after receiving
the hint. However, it also has limitations. The total percentage of subjects
producing the analogous solution is not as pure a measure of potential
application as would be provided by a separate group of subjects who
were given a hint on their first encounter with the radiation problem.
Subjects who do not immediately notice the relevance of the prior analog
may tend to forget it during their initial attempts to solve the target problem. In addition, their initial attempts may produce a “set” effect thatinterferes with later application of the analogy. Our within-subject procedure may thus make it difficult for a condition that produces a relatively
low rate of initial noticing to later “catch-up” and achieve an equal rate of
total application. We should note, however, that Gick and Holyoak (1980)
used both between-subject and within-subject procedures in different experiments and obtained essentially the same estimates of the percentage
of subjects who could apply the analogy (compare their Experiments II
and V). There is therefore reason to think the potential bias associatedwith a within-subject procedure is not a serious one, at least for theradiation problem; nevertheless, it necessitates a degree of caution ininterpreting certain of our results.
We should also note that the present study does not include any control
groups that attempted to solve the radiation problem without receiving astory analog. Gick and Holyoak (1980) tested a variety of such control
groups, which received either an irrelevant story that was not analogous
to the radiation problem or no story at all; no more than 10% of such
subjects ever produced the convergence solution. Because the present
experiments used subjects drawn from essentially the same subject
population, we assumed a base rate of 10% for producing the criticalsolution without an analogy.
The experimental manipulations used in the present study involve variations in the processing of the prior analog, rather than of the target
problem. The target problem is explicitly presented as a “problem”; the

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HOLYOAKsubject can try out alternative representations of it in the course of thesolution attempt. The analogous story, however, has been encoded inmemory prior to presentation of the target problem. It is presented as a“story” rather than as a “problem”; subjects are never given any impetus
to search for alternative representations of it, nor are they led to expect
the story to later serve as the basis for a potential analogy. More generally, the analogist is free to mentally transform the representation of asituation currently being considered; but the memory representations ofpotential analogs will presumably have been “fixed” at the times of their
original encodings and cannot be processed further until they have been
retrieved and their pertinence noticed. The later accessibility of an analog
should therefore be especially dependent on the nature of its initial encoding. Accordingly, all of the present experiments involve manipulationsof the encodings of story analogs that are presented prior to the target
problem.
Manipulationsof Schema Induction

All of the present experiments investigated the effects of manipulations
that might influence the induction of a problem schema. One major manipulation, which serves to organize the experiments, follows directly fromour theoretical analysis. This basic variation is extremely simple-we
presented subjects with either one prior story analog or else two. Ourearlier analysis of schema induction suggests that this distinction is potentially crucial. A schema can be formed by abstracting the mapped identities common to two analogs. Such a mechanism clearly hinges on theprovision of two prior analogs, which can be mapped together to induce
the schema. Indeed, the schema is defined by the correspondences between two analogs. Accordingly, provision of two analogs should foster
induction of an independent schema, and hence facilitate subsequenttransfer.
Nevertheless, the schema will be implicit within the representation of
an individual analog. It is therefore conceivable that a schema could be
abstracted in whole or in part from a single example. What would be
required is some mechanism, other than mapping with a further example,
which could make the schema explicit. Part I of the empirical section
below includes three experiments that explored encoding manipulations
that might promote schema induction from a single analog. The manipulations used were summarization of the prior analog (as opposed to roterecall), provision of a verbal statement of the implicit schema as part ofthe story, and provision of a diagrammatic representation of the schema.
Part II includes three further experiments in which two prior analogs were
presented. Two of the latter experiments also investigated the effects ofprovision of a verbal or diagrammatic representation of the schema.

ANALOGICALPART I: REASONING

TRANSFER

13

FROM A SINGLE ANALOG

Experiment 1
As noted above, Gick and Holyoak (1980) presented an initial story
analogy to subjects in the guise of a recall experiment. It could be argued
that the type of encoding encouraged by recall instructions may be particularly ill-suited for producing subsequent transfer to an analogous problem. Because the instructions emphasized verbatim recall, subjects may
have refrained from doing any kind of abstraction that might have highlighted the convergence schema implicit in the story. Perhaps subjects
would be more likely to abstract the underlying solution principle if they
were required to summarize the story, rather than to recall it. The notionthat summarization might foster schema abstraction acquires a degree ofplausibility from evidence that summaries, as compared with recall protocols, typically include a proportionately greater number of propositions
drawn from higher levels of a representational hierarchy (Thorndyke,
1977; Kintsch & van Dijk, 1978).
Gick (Note 1) compared transfer from a story to the radiation problem
when the story was presented in the context of either a recall or summarization task. She found no firm evidence that summarization instructions
facilitated either abstraction of a convergence schema or subsequenttransfer. Experiment 1 investigated this same issue using a different
transfer problem (since the exclusive use of the radiation problem is an
obvious limitation of our previous research).
Method. In choosing a new target problem we followed a conservative tack, selecting aproblem that shared many of the major properties of the radiation problem. In particular, thenew problem was also spatial in its nature, did not tax working memory, and allowed avariety of potentia1 solution plans. Our choice was Maier’s (1930, 1931) “cord problem”:
Suppose you are in a room, where two cords are hung from the ceiling. The two cords areof such a length that when you hold one cord in either hand, you cannot reach the other.
Your task is to tie the ends of these cords together. To help you in this task, you may use anyof the objects listed below, which are also in this room: poles, clamps, pliers, extension
cords, tables, chairs.
Maier distinguished among several types of solutions to this problem, of which he wasmainly interested in the “pendulum”
solution. This is to tie a weight to one cord and swing it
so it becomes a pendulum. The other cord can then be brought toward the center, theswinging cord can be caught as it approaches the midpoint, and then the two cords can betied together. Maier found that 39% of his subjects produced the pendulum solution without
any hints from the experimenter.
This suggests that in the absence of an analogy, thependulum solution to the cord problem is easier to discover than is the convergence solution
to the radiation problem.
To provide a potential analogy, we wrote a story called “The Birthday Party” (seeAppendix I), in which two ribbons are tied together after a pair of scissors is used to turn oneof them into a pendulum. The story was embellished with various details more or lessirrelevant to its problem-solving
component. If summarization instructions lead subjects tofocus on the problem-solving
aspects of the story, the presence of extra details should make

14

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HOLYOAKit more likely that summarization instructions will improve transfer performance relative torecall instructions.
The experiment was administered in booklet form to high school classes (grades 10 to 12)
in Bloomtield, Michigan. Subjects were divided into three conditions: analog recall, analog
summary, and control summary. The analog conditions either recalled or summarized thestory analog prior to attempting the cord problem. The control-summary condition served toestablish the base rate for production of the pendulum solution in the absence of an analogy.
Subjects in the latter condition received a story that was not analogous to the cord problem(“The Wine Merchants”; see Appendix IV in Gick & Holyoak, 1980);in other respects theyreceived the same treatment as did those in the analog-summary condition.
Subjects were told they were participating in a two-part experiment, involving story
comprehension and problem solving. Before distribution of the stories, subjects were told
either that they would have to summarize or else recall the story later, so they should pay
attention to the major points of the story (summary conditions), or else learn it in as muchdetail as possible (analog-recall condition). Following the 3-min period during which theystudied the story, they were given detailed instructions about how to either summarize orrecall it. After the protocols were written, the cord problem was distributed and subjects
were required to produce as many solutions as possible. A questionnaire that queried subjects about their noticing of the story’s relevance, the solution that was suggested by thestory, and their prior familiarity with this solution was then distributed. Sixty-nine subjects
served in the experiment: 25 in the analog-recall condition, 24 in the analog-summary condition, and 20 in the control-summary condition.

Results and Discussion. Table 2 presents the percentage of subjects ineach condition who produced the pendulum solution to the cord problem
before the hint, after the hint, and in total. Maximum likelihood chi
squares (G2) were calculated for the frequency data (Bishop, Fienberg, &Holland, 1975; Hays, 1973, p. 737, Eq. 17.8.2). The facilitative effect of aprior story analog was at least as striking as it had been in our earlier
studies using the radiation problem. Combining the two analog conditions, subjects who received the analogous story produced more pendulum solutions prior to the hint than did those who received the control
story (71 vs 20%), G2(1) = 15.8,~ < .OOl. The advantage of the analog
conditions was maintained in a comparison of total solution frequencies(98 vs 30%), G2(1) = 30.1, p < .OOl. However, the two analog conditions,recall and summary, did not differ significantly either in solutions prior to

TABLE 2
Percentage of Subjects Producing Pendulum Solution
to Cord Problem (Experiment 1)

Analog recall
Analog summary
Control summary

Before
hint

After
hint

68 (17)25 (18)20 (4)

25 (6)
10 (2)

Note. Frequencies are given in parentheses.

29 (7)Total

N

96 (24)100 (24)
30 (6)

25
24
20

ANALOGICAL

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15

the hint (68 vs 75%), G2(1) < 1, or in total solution frequencies (96 vs100%).
The results of Experiment 1 thus yielded no evidence that summarization of the analog facilitates either noticing or applying an analogy morethan does recall. While the lack of a difference in total solution frequencies could obviously be attributable to a ceiling effect, the absence of areliable summarization advantage in initial noticing cannot. As in thecomparable experiment using the radiation problem (Gick, Note l),scrutiny of the story protocols revealed few differences between summaries and recall protocols, except that the summaries were shorter.
Neither experiment yielded any clear indication that instructions to summarize are more likely to foster either abstraction of an explicit problem
schema or subsequent analogical transfer. Encoding for recall may often
involve a degree of abstraction similar to that associated with encoding forsummarization, particularly when study time is limited, as in the present
experiment. For example, the Kintsch and van Dijk (1978) model suggests that multiple levels of macrostructure are derived in either case;summaries may be briefer than recall protocols due to differences in output processes, rather than to differences in the internal representations of
the story.
The “gap” between initial noticing and eventual application of theanalogy was much smaller in the present experiment with the cord problem than in prior experiments with the radiation problem. Because theexperimental situations differed in many ways (e.g., the cord problem issomewhat easier to solve in the absence of an analogy), strong conclusions are unwarranted. But while we do not have a formal metric ofanalogical similarity with which to compare such diverse analogies as themilitary and radiation problems versus the ribbon and cord problems, it isintuitively compelling that the analog is semantically closer to the target
problem in the latter case. For example, ribbons and cords are much moresimilar than are armies and rays; both pendulum solutions accomplish agoal of “tying,” while the analogous convergence solutions serve either
to “capture” or to “destroy.” In terms of our earlier theoretical analysis,
the former mapping seems to be characterized by a greater degree ofidentity relative to difference. The present results are thus consistent with
our prediction that more similar analogs will have a greater number ofpotential retrieval cues to link them, so that the prior analog will more
often be noticed without an explicit hint.
Experiment2

The failure of summarization instructions to foster abstraction of aproblem schema from a story analog is not surprising, given the nondirec-

16

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HOLYOAKtive nature of the manipulation. Experiment 2 introduced what would
seem to be a more direct approach: the story analog was augmented by an
explicit verbal statement of the principle underlying the implicit problem
schema. Kohler (1940) found that explaining the abstract principle uponwhich the solution to an algebra problem depended sometimes facilitatedanalogical transfer more than did provision of the initial example alone.
There was therefore some reason to expect the addition of a verbal principle to be of benefit in our paradigm.
Method. The basic story analog used in Experiment 2 was “The General” (see Appendix II), with the radiation problem as the transfer task. Subjects served in one of three
conditions, which differed only in the material initially studied. Those in the analog-only
condition read “The General”; those in the analog-plus-principle condition read the same
story, except that the following sentence was added as a final paragraph: “The genera1
attributed his success to an important principle: If you need a large force to accomplish some
purpose, but are prevented from applying such a force directly, many smaller forces applied
simultaneously from different directions may work just as well.” This verbal principle wasmeant to be an explicit statement of the essential aspects of the schema for convergenceproblems. Finally, subjects in the principle-alone condition did not receive the story, but
only the bare statement of the principle (“If you need. . “). If the verbal statement caneffectively convey the convergence schema to subjects, even without an accompanying
specific example, then subjects in the latter condition should produce the convergencesolution to the radiation problem more frequently than the base rate.
Subjects were tested in small groups. Those in the analog-only and analog-plus-principle
conditions were told they were first to learn a story: they then were allowed 3 min to read thestory, after which they were asked to recall it. Subjects in the principle-only condition were
given a slightly different task. They were told the first part of the experiment involved“selecting statements to use in a study of comprehension” and that they were to first read astatement and prepare to paraphrase it. After studying the statement for 3 min, they wereasked to paraphrase it and to rate its plausibility and comprehensibility. After their initialtask was completed, subjects in all conditions attempted to solve the radiation problem inthe usual two-pass manner: first without a hint to use the prior story or statement, and thenwith such a hint.
Eighty-eight subjects participated in the experiment, distributed approximately equally
across the three conditions.
Results and Discussion. Table 3 presents the percentage of subjects ineach of the three conditions who produced the convergence solution before the hint, after it, and in total. The results were straightforward:
solution frequencies did not differ significantly among the conditions,
TABLE 3
Percentage of Subjects Producing Convergence Solution(Experiment 2)

Analog plus principle
Analog only
Principle only

Before
hint

After
hint

Total

N

32 (10)48 (15)50 (14)38 (11)80 (25)79 (22)66 (19)3128
29

29 (8128 (8)

Note. Frequencies are given in parentheses.

ANALOGICAL

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17

either before the hint, G2(2) < 1, or in total, G2(2) = 2.05, p = .36. In
particular, the performance of subjects in the analog-plus-principle condition was virtually identical to that of those in the analog-only condition:
about 30% produced the convergence solution without a hint, and about
80% produced it in total. These figures are comparable to those obtainedin previous experiments that used the military story and radiation problem. Experiment 2 thus yielded no support for the hypothesis that augmenting the story analog with a verbal principle would increase analogicaltransfer.
This does not mean, however, that the verbal principle was totallyineffective, because a substantial proportion of subjects in the principleonly condition also produced the convergence solution. In all three conditions (considered separately) the 95% confidence limits for the percentage of such successful subjects excluded the base rate of lo%, even prior
to the hint. It is therefore clear that the information in the verbal statement was often sufficient to allow generation of the convergence solution
to the transfer problem; however, once the subject received a specificstory analog, addition of the abstract principle conveyed no further benefit.
It therefore seems that the verbal principle did not foster abstraction of
the convergence schema from the analog. One possibility is that subjects
who received the story simply ignored the concluding statement. However, inspection of the recall protocols for the analog-plus-principle subjects suggested otherwise. All but three of the 31 subjects included at least
one aspect of the abstract statement of the convergence principle in theirrecall of the story. It is possible, however, that subjects did not attend
carefully to the relationship between the final statement and the concrete
instantiation of a convergence solution embedded in the story. That is,they may not have mapped the story and the statement to abstract theconvergence schema implicit in the story itself. This seems particularly
plausible given that they were simply asked to learn the story and did not
expect to have to apply the information in a subsequent problem-solving
task. The present findings therefore should not be taken as evidence that averbal principle can never aid in clarifying an example. Augmenting an
example with a principle may be beneficial when the information is presented in an explicit problem-solving context (Kohler, 1940), rather than in
an incidental task. However, our results indicate that simply adding a
principle to a specific analog by no means guarantees that an abstractschema will be abstracted.
Experiment

3

Although the problems used in our experiments have always beenstated verbally, they are fundamentally spatial in nature. Perhaps an
abstract principle would better serve to highlight the convergence schema

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GICK AND HOLYOAK

implicit in a single analog if the principle were presented in a spatial rather
than a verbal mode. To test this possibility, Experiment 3 used a design
analogous to that of Experiment 2, but with visual diagrams instead of averbal statement of the principle.
Method. Subjects again served in one of three conditions. Those in the analog-only
condition read the same military story as had been used in the comparable condition ofExperiment 2. Subjects in the analog-plus-diagrams condition also read the same story, but
modified slightly to refer to a pair of accompanying diagrams, depicted in Fig. 1. Diagram A,a single large arrow, represents the desirable but blocked plan of sending a large force from asingle direction. The story referred to this diagram as an illustration of the general’s initialplan “to launch a full-scale direct attack.” Diagram B, several smaller converging arrows,
represents the alternative, successful plan of sending small forces from multiple directions.
The story referred to the latter diagram as an illustration of the general’s solution in a
sentence inserted immediately before the last sentence of the story (see Appendix II). These
simple arrow diagrams were intended to represent the convergence principle at an abstractlevel, not necessarily tied to the story’s specific military context.
Subjects in both of the above conditions were told to pay attention to the major points ofthe story during their 3-min study period and that they would later have to summarize it.
Those in the analog-plus-diagrams condition were told in addition to pay attention to thediagrams and to incorporate them into their summaries. After the study period subjects inboth analog conditions were asked to write a summary and to rate the story’s plausibility andcomprehensibility. Subjects in a third condition, diagrams only, received the diagrams without any accompanying story. They were told that the first part of the experiment involved
pattern recognition; they were to study the diagrams for 3 min so that they could later
reproduce them. These subjects then drew the diagrams from memory and rated theirperceptual complexity and the difficulty of reproducing them.
After completing their initial task, all subjects received the radiation problem and finalquestionnaire in the usual two-pass manner. The final questionnaire for the diagrams:only
condition asked subjects to give a solution suggested by the prior diagrams. Sixty-six subjects served in the experiment: 26 in the analog-plus diagrams condition, 25 in the analogonly condition, and 15 in the diagrams-only condition.

Results and Discussion. Table 4 presents the percentage of subjects ineach condition who produced the convergence solution to the radiation
problem. There was no indication that spatial diagrams, any more than theverbal statement used in Experiment 2, constituted an effective addition
to the basic story analog. The percentage of subjects producing the convergence solution prior to the hint was actually lower in the analog-plusdiagrams condition than in the analog-only condition, although not signifi-

FIG. 1. The two diagrams used in Experiments 3 and 6 to illustrate the principle underlying the convergence solution.

ANALOGICAL

19

TRANSFERTABLE 4
Percentage of Subjects Producing Convergence Solution(Experiment 3)

Analog plus diagrams
Analog only
Diagrams only

Before
hint

After
hint

Total

N

23 (6)

58 (15)36 (9)
60 (9)

81 (21)76 (19)67 (10)2625
1540 (IO)
7 (1)Note. Frequencies are given in parentheses.

cantly so, G2(1) = 1.71, p = .19. The total solution frequencies in bothconditions were very similar (approximately SO%),G2(1) < 1.
Although both analog conditions yielded results much like those inprevious comparable experiments, the diagrams-only condition provided
an interesting contrast. Combining the two analog conditions, 3 1% of theirsubjects produced the convergence solution without the hint, as compared to only one of the 15 diagrams-only subjects (7%). The lattersubject reported solving the radiation problem without thinking of the
diagrams. However, nine additional diagrams-only subjects generated thetarget solution once the hint was given; the total for that condition, 67%,
was not significantly less than the figure of 78% obtained for the analog
conditions, G2(1) < 1. The diagrams-only condition thus yielded the moststriking discrepancy between initial noticing and eventual application we
have yet observed: information in memory, sufficient to trigger the convergence solution for most subjects, was literally never spontaneously
noticed.
This result in fact supports our semantic retrieval-cue analysis of noticing analogies. When the diagrams were presented alone in the context
of a “pattern recognition” task, it is unlikely that subjects assigned any
semantic interpretation to them, and surely none even approximating theconvergence schema. As a result, no semantic retrieval cues were subsequently available to link the radiation problem with the diagrams. But
once subjects had been explicitly told to consider the prior diagrams, they
were able to interpret them by means of a mapping process. For example,
they may have noticed the abstract similarity of the large unidirectional
arrow to the desirable but blocked possibility of directing high-intensity
rays at the tumor. Once the initial arrow had been interpreted as a representation of the intensity and direction of rays, the relationship between
the two diagrams could be construed as a transformation of one large
unidirectional ray into several small converging ones, thus illustrating theconvergence solution. Such a chain of reasoning might be termed“analogical bootstrapping.” The antecedent conditions given in thestatement of the transfer problem are first used to interpret the

20

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HOLYOAK“antecedent” diagram; once the components of the first diagram havebeen interpreted by constructing an initial mapping, the “consequence”
represented by the second diagram can be decoded.
But if the diagrams can indeed be interpreted as an abstract analog to
the radiation problem, why did they not facilitate analogical transfer whenpresented in the context of the military story? They certainly were notignored; every subject reproduced them accurately after the study period.
One might have expected the diagrams to highlight the abstract convergence schema implicit in the story. But as was the case with the verbal
statement used in Experiment 2, scrutiny of subjects’ story protocols
indicated that nothing of the sort actually occurred. Except for references
to the diagrams, there were no obvious differences between the summaries written by subjects in the analog-plus-diagrams conditions andthose written by subjects in the analog-only condition. In neither case wasthere anything in the protocols to suggest a more abstract interpretation of
the general’s solution strategy, which might apply to disparate problems.
In fact, there were signs that rather than the diagrams providing a more
abstract interpretation of the story, the story provided a more specificinterpretation of the diagrams. Two subjects stated that the general divided his men into six groups, which were dispatched to six roads, “asillustrated in the diagram.” The story does not say how many roads wereused, but the second diagram indeed shows six arrows. It seems that in
the story context the diagrams were simply taken to represent roads,
rather than abstract directions of force. Given that the diagrams did notencourage abstraction of the convergence schema from the story analogy,
their failure to increase analogical transfer is entirely in accord with ourearlier analysis.
PART II: FORMATIONAND USE OF SCHEMAS

We have examined three manipulations-summarization
instructions,addition of a verbal statement of the underlying principle, and addition ofa diagrammatic representation of it-that were designed to promotespontaneous transfer by making the schema latent in the story analog
more explicit. These all failed to facilitate transfer from a single analog.
Furthermore, none of these manipulations appeared to foster abstractionof a general problem schema from one example, even though the verbal
statement amounted to an explicit statement of such a schema, and boththe verbal statement and the diagram produced transfer when presented
alone (although in the latter case only with a hint). A common limitation of
all these procedures is that they do not allow induction of a schema by
mapping, since only one example was provided. The experiments in Part
II investigated the induction of a problem schema from multiple analogs.

ANALOGICAL

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21

Experiment 4
Experiment 4 explicitly compared the effectiveness of two vs one prioranalogs in producing analogical transfer. In addition, the experimental
procedure gave subjects an opportunity to write down a description that
could approximate the convergence schema. The quality of their schema
descriptions could then be related to their subsequent performance insolving the transfer problem.
As well as simply providing a second analog, we varied whether the twostory analogs were drawn from similar or dissimilar domains (where theanalog domains were always dissimilar to the medical domain of the radiation problem). There is reason to expect the effect of analog similarity to
be complex. Studies of perceptual category learning have often found thattraining with relatively diverse examples leads to superior transfer performance (Posner & Keele, 1968; Fried & Holyoak, Note 3); comparable
effects have been demonstrated with artificial semantic concepts (Nitsch,
Note 4). On the other hand, perceptual categories are learned more slowly
when the training exemplars are highly variable (Fried & Holyoak, Note3); and when the number of initial exemplars is very small, high variability
can actually impair transfer performance (Peterson, Meagher, Chait, &Gillie, 1973; Homa & Vosburgh, 1976).
The general picture appears to be that dissimilar training examples
make it more difficult to learn a concept, but allow it to be used more
flexibly once acquired. A comparable pattern might be expected in thepresent paradigm. In terms of our earlier analysis, the optimal schema for
a set of analogs will be a representation that captures the essential relations relevant to the solution, while excluding irrelevant domain-specificdetails. The schema that can be abstracted from two analogs will dependon the set of mapped identities they share. Because the optimal-convergence schema is quite abstract, it would seem that the mapped identities linking two relatively dissimilar analogs would more likely correspond to an optimal schema (because they are similar only at an abstractlevel). A schema induced from two similar examples would often include
information too specific to apply to a disparate transfer problem. However, the mapping process may be substantially more difficult to accomplish if the analogs are dissimilar, because the differences will bemany and the identities few and abstract. In contrast, it will be relatively
easy to map two similar analogs, which will afford salient and specificpoints of correspondence. Thus, while a mapping between dissimilar
analogs may potentially yield relatively optimal schemas, the actual
schema that results may fall short of that which emerges from a mapping
between two similar analogs.
Method. Four story analogs to the radiation problem were used (see Appendix II). One
was“The General” (by now a veteran of many campaigns in our previous experiments); the

22

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HOLYOAKother three were written for this experiment. “The Commander,” a story involving an attack
on an island using multiple bridges, provided a second military analog. The other pair ofstories shared a fire-fighting theme. “Red Adair” had the resourceful hero extinguish an
oil-well fire by using multiple hoses; “The Fire Chief” accomplished a similar feat with theaid of a bucket brigade.
All subjects initially read two stories. Subjects were divided into three basic groups, eachof which received a different type of story pair. Those in the two-similar-analogs condition
received either the two military stories or the two fire-fighting stories, while those in thetwo-dissimilar-analogs condition received one of each. Subjects in the analog-plus-control
condition received just one story analog, plus a disanalogous control story (“The IdenticalTwins”) that had proved useless as a prompt for the convergence solution in earlier experiments (Gick & Holyoak, 1980). Across different subjects, all four story analogs were used
about equally often within each of these three conditions, and the order of the stories wascounterbalanced.
Subjects were told that the first part of the experiment involved screening some stories foruse in a study of story comprehension. They were asked to read each story in the given orderand to write brief summaries of each, with the stories still available for reference. Subjects
were allowed to keep the stories while answering questions about them in order to facilitatecomparison of one to the other. The subjects also rated the comprehensibility of each story.
Then, in what may be the most critical aspect of the procedure, subjects were asked todescribe in writing, as clearly as possible, ways in which the stories were similar. This task
was intended to elicit a mapping between one story and the other, and hence to potentiallylead to abstraction of a convergence schema. Furthermore, the written descriptions could
later be scored for presence and quality of the schema. Finally, subjects were asked to rate
the similarity of the two stories on a 7-point scale. The radiation problem was then presented
in the usual two-pass manner, except that the final questionnaire asked subjects to write
down a solution suggested by either one or both of the prior stories.
A total of 98 subjects were tested: 28 in the two-similar-analogs condition, 23 in thetwo-dissimilar-analogs condition, and 47 in the analog-plus-control condition. Subjects in thelatter condition were divided into four approximately equal subgroups, each of which read a
different story analog, plus the irrelevant story.

Results and Discussion. Table 5 presents the percentage of subjects ineach condition who produced the convergence solution prior to the hint,
after it, and in total. While there was a small trend toward a greater
frequency of unprompted solutions when the two prior analogs were dissimilar rather than similar (52 vs 39%), this difference did not approachsignificance, Gz( 1) < 1. Because these two conditions that received two
analogs also did not differ in total solution frequencies, they were combined in order to make a comparison with the one-analog-plus-control
TABLE 5Percentage of Subjects Producing Convergence Solution(Experiment 4)

Two similar analogs
Two dissimilar analogs
One analog plus control

Before
hint

After
hint

Total

N

39 (11)52 (12)21 (10)39 (11)31 (7)
32 (15)79 (22)
83 (19)53 (25)28
2347

No@. Frequencies are given in parentheses.

ANALOGICAL

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condition. This comparison yielded clear differences. Forty-five percent
of subjects in the two-analog conditions generated the convergence solution prior to the hint, as opposed to only 21% of subjects in the one-analog
condition, G2(1) = 6.35, p < .Ol. A comparable advantage for the twoanalog conditions was obtained when total solution frequencies werecompared (80 vs 53%), G2(1) = 8.36,~ < .003. It was clearly the case that
two analogs were better than one.
We next sought to assess the relationship between quality of theschemas induced by subjects, as evidenced by their descriptions of thesimilarities between the two stories, and analogical transfer. This analysis
was performed only for the two-analog conditions. The descriptions werecategorized into three levels of schema quality. In order for a description
to qualify as a “good” schema, the basic idea of having forces converge
from different directions had to be present either explicitly or as an inference. (The idea of using different directions was inferred if the subject
mentioned the simultaneous use of multiple forces.) In addition, at least
one other major aspect of the analogy had to be expressed: either the useof multiple small forces or the parallels in the initial problem situations(e.g., centrally located targets). An example of a good schema is: “Bothstories used the same concept to solve a problem, which was to use manysmall forces applied together to add up to one large force necessary todestroy the object.” An “intermediate” schema contained only one ofthese major features; e.g., “In both cases many small forces were used.”“Poor” schemas contained none of the basic aspects of the convergence
principle. They usually either alluded to a similarity between the solutions
that was abstract to the point of vacuity (e.g., “In both stories a problem
was solved using logical means”) or did not focus on the problem-solving
aspects of the stories at all (e.g., “In both stories a hero was rewarded for
his efforts”). The descriptions were evaluated by two independent judges,
and disagreements, which were infrequent, were resolved by discussion.
The schema analysis revealed no significant differences as a function ofsimilarity of the two analogs; nor did the relationship between schema
quality and transfer performance vary reliably across the two conditions(although the cell frequencies were too low for the latter null result to betaken at all seriously). Combining both conditions, 21% of subjects produced good schemas, 20% produced intermediate schemas, and the
majority, 59%, produced poor schemas. The question of central interest iswhether schema quality was at all predictive of success in generating theconvergence solution to the radiation problem. As the data in Table 6
indicate, such a relationship is strikingly evident. The cell frequencies
presented in Table 6 differ reliably, G2(4) = 15.8,~ < .005, and the patternof difference is quite simple: the better the schema, the more successful
was transfer performance. Thus, fully 91% of subjects scored as having
good schemas generated the convergence solution Mithout a hint; none

24

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HOLYOAK
6

Percentage of Subjects Producing Convergence Solution at Each Stage
as a Function of Schema Quality (Experiment 4)
Timing of convergence solution
Schema Quality

Before
hint

After
hint

Not
produced

N

Good
IntermediatePoor

91 (10)40 (4)
30 (9)9 (1)
50 (5)
40 (12)0 (0)10 (1)
30 (9)

11
10
30

Note.

Frequencies are given in parentheses.

failed to produce it eventually. Forty percent of subjects with intermediate schemas produced the target solution without a hint, and 90%
produced it in total. The comparable figures for those with poor schemas
were 30% unprompted and 70% in total. To state the results in different
terms, 90% of the subjects who failed to ever produce the convergencesolution were scored as having poor schemas.
One must be cautious in suggesting a causal basis for this strong relationship between schema quality and transfer performance; it may bethat people who are skillful problem solvers write good schemas and also
apply analogies well. However, our subsequent experiments will lend a
degree of support for a causal interpretation. The observed pattern iscertainly consistent with the view that the task of evaluating story similarities invokes a mapping process which may lead to the incidental
abstraction of a problem schema; the more closely the induced schema
approximates the optimal convergence schema, the more likely the person is to notice its relevance to the subsequent transfer problem and to beable to derive the appropriate solution.
The schema analysis also suggests explanations of what would otherwise seem to be puzzling aspects of the performance levels observed in
the present transfer task. First, the two-analog conditions yielded solution
frequencies of 45% without a hint and 80% in total. At least the latter
figure is not remarkably different from figures obtained in previous experiments with a single prior analog, which might seem to contradict our
prediction that reasoning from a schema will be more effective than reasoning from an analog. However, the schema analysis revealed that virtually 60% of the subjects in the two-analog conditions produced poor
schemas, which is to say no part at all of the convergence schema. Thosesubjects who wrote intermediate or good schemas indeed outperformed
any of our previous one-analog conditions.
A second and related puzzle is why the present one-analog condition
did so poorly: 21% solutions without a hint, 53% in total, as opposed tothe usual figures of roughly 30 and 75%. Although three of the four story

ANALOGICAL

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25

analogies had not been used in previous experiments, the new stories
were no less effective than the old one. However, note that the present
one-analog-plus-control condition actually involved two stories, only oneof which was an analogy to the radiation problem; furthermore, subjects
in this condition also described the similarities between the two stories,
just as did those in the two-analog conditions. The procedure thus encouraged analog-plus-control subjects to also abstract some sort ofschema linking their two stories. But because the two stories were totallydisanalogous, except that both described problem situations, the resulting
schemas were invariably poor-indeed, very poor-approximations
tothe convergence schema. For these subjects, the induced schema may
have actually interfered with their ability to later retrieve and apply theactual analog during the transfer task.
Experiment 5
Experiment 4 yielded clear evidence that a problem schema can be
induced by mapping two analogs and that the quality of that schema
predicts subsequent transfer performance. However, only about 40% of
the subjects described the similarities between the two analogs in terms
that approximated the convergence schema. Our analysis predicts thatany manipulation that can facilitate schema formation will boost
analogical transfer. Once the subject has been given two analogs, themapping process should be aided by any device that calls attention to thecritical correspondences. We have already introduced two such devices inearlier experiments: a verbal statement of the convergence principle and adiagrammatic representation of it. Neither manipulation facilitatedschema abstraction from a single analog. However, these negative results
may simply reflect the fact that a mapping process requires two analogs inorder to operate at all. Once this prerequisite is met, highlighting devices
should have a positive effect. Experiment 5 therefore examined the effectof adding the verbal statement, used previously in Experiment 2, to twoprior analogs.
kfethod. To simplify the experimental design only one pair of dissimilar story analogs
were used: “The General” and “The Fire Chief.” Subjects served in one of two conditions.
Those in the without-principle condition read the two stories just as they appear in Appendix
II. Those in the with-principle condition read the identical stories, except that the verbal
statement of the convergence principle, previously used in Experiment 2, was appended as
the final paragraph of each. The statement was worded exactly as it had been in the earlier
experiment (except, of course, that the corresponding sentence for “The Fire Chief” began,“The tire chief attributed his success. . .“). The statement was thus designed to focus the
subjects’ attention on the critical aspects of the schema implicit in each of the two analogs.
The two-analog procedure of Experiment 4 was modified slightly so as to be more comparable with that of Experiment 2. Subjects were first told to study the two stories carefully for
5 min in preparation for answering questions about them. The stories were then collected,
and the remainder of the initial story task was done from memory. Subjects were asked tobriefly summarize each story, rate the comprehensibility of each, describe as clearly as

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HOLYOAKpossible the ways in which the situations in the two stories seemed similar, and rate theiroverall similarity. After this initial task was completed, the radiation problem was administered in the usual two-pass manner.
Subjects were tested in several different classrooms. A total of 143 subjects were divided
about evenly between the two conditions.

Results and Discussion. In sharp contrast to the comparable one-analog
study (Experiment 2), addition of the verbal statement had a clear positiveeffect on transfer from a pair of analogs. As is shown in Table 7A, 62% of
the subjects in the with-principle condition produced the convergencesolution without a hint, as compared to only 40% of those in the withoutprinciple condition, G2(1) = 6.75,~ < .Ol. The frequency of noticing in theformer condition is in fact higher than that obtained in any of our previous
experiments with the radiation problem. The with-principle condition
maintained a slightly smaller advantage in total solution frequency (82 vs67%), G2(1) = 4.34,~ < .05.
Subjects’ similarity descriptions were scored for schema quality in essentially the same manner as in Experiment 4. A few subjects in the
with-principle condition simply said that the two stories illustrated thesame principle; such descriptions were counted as good schemas only if
the principle was clearly stated in their summary of each story. As thedata in Table 7B indicate, addition of the principle had a strong influenceon schema quality, G2(2) = 28.4,~ < .OOl. Sixty percent of subjects in the
without-principle condition wrote poor schemas, just as in the comparable
condition of Experiment 4; only 10% gave good schemas. In contrast,
only 23% of those in the with-principle condition produced poor schemas,
whereas 44% produced schemas scored as good.
TABLE 7
A. Percentage of Subjects Producing Convergence
Solution (Experiment 5)

With principle
Without principle

Before
hint

After
hint

Total

N

62 (45)40 (28)20 (15)27 (19)82 (60)67 (47)1370

B. Percentage of Subjects at Each Levelof Schema Quality
Schema quality

With principle
Without principle

Good

Intermediate

Poor

N

44 (32)10 (7)

33 (24)30 (21)23 (17)60 (42)73
70

Note. Frequencies are given in parentheses.

ANALOGICAL

27

TRANSFERTABLE 8Percentage of Subjects Producing Convergence Solution at Each Stage
as a Function of Schema Quality (Experiment 5)
Timing of convergence solution
Schema Quality

Before
hint

After
hint

With principle
Good
IntermediatePoor

15 (24)58 (14)41 (7)

19 (6)
29 (7)

Without principle
Good
IntermediatePoor

57 (4)
57 (12)29 (12)29 G-912(2)24 (5)
29 (12)

Not
produced

6 (2)N

13 (3)
47 (8)

32
24
17

14 (1)
19 (4)
43 (18)72142Note. Frequencies are given in parentheses.

Table 8 presents data regarding the relationship between schema quality and transfer performance. Data for the two experimental conditions
were tabulated and analyzed separately. For the with-principle condition,
schema quality had a strong influence on generation of the convergencesolution, G2(4) = 13.0, p < .02. For example, fully 94% of the subjects
who wrote good schemas eventually produced the solution, as opposed toonly 53% of those who wrote poor schemas. The overall relationshipbetween schema quality and solution generation was less robust for the
without-principle condition, G2(4) = 7.00,~ = .14, due to the lack of clear
differences between subjects who gave good versus intermediate
schemas. However, little weight can be attached to this null finding, because only seven subjects in this condition wrote good schemas. Whensubjects with good and intermediate schemas were combined, schema
quality had a significant positive effect for the without-principle condition
as well, G2(2) = 6.89,~ < .05. Eighty-six percent of the subjects who gave
descriptions relevant to the convergence principle eventually produced
the target solution, as opposed to only 57% of those who gave poor
schemas.
Experiment

6

The results of Experiment 5 again provided clear evidence of the formation and use of schemas in analogical problem solving; furthermore,they demonstrated that a verbal statement of the solution principle can aid
in the induction of a schema from two analogs and substantially improve
both noticing and application of the information in a subsequent transfer
task. Experiment 6 was designed to seek comparable effects of a visual

28

GICK

AND

HOLYOAKrepresentation of the principle. The diagrams previously used in Experiment 3, where they failed to facilitate abstraction of a schema from one
analog, were now presented along with two analogs. In addition, Experiment 6 obtained further evidence regarding the influence of similarity
between the two prior analogs, which had yielded no clear effects inExperiment 4.
As we argued in connection with Experiment 4, there are reasons to
expect variations in similarity of the analogs to have complex and sometimes offsetting effects. We would expect provision of diagrams or otherdevices that highlight the problem schema implicit in each analog to foster
induction of the schema from either similar or dissimilar analogs, but for
somewhat different reasons. The danger with similar analogs is that the
mapping process will yield a set of mapped identities that includes
domain-specific details irrelevant to the more general schema. The diagrams in Fig. 1 should reduce this problem, by focusing the subject’s
attention on those central correspondences causally related to the analogous solution plans. The diffkulty raised by a pair of dissimilar analogs isessentially the converse: the analogical correspondences will be few andrelatively abstract, so that the mapping process may simply fail to findthem. In this case the diagrams should also prove helpful, because they
will serve to isolate the essential correspondences which they abstractly
represent.
Method. Subjects served in one of four basic conditions, all of which received twostory analogies. The four stories used in Experiment 4 were again used in Experiment 6.
Subjects read either two similar analogs (both from either the military or the fire-fighting
domain) or else two dissimilar analogs (one from each domain). The assignment of stories toconditions, and the order of the two stories in a pair, were fully counterbalanced, so thatacross subjects each story was used about equally often in similar and dissimilar pairs.
Subjects in the without-diagrams conditions read the stories as they are presented in Appendix II. Those in the with-diagrams conditions received a separate sheet of paper with the twoarrow diagrams illustrated in Fig. 1. The stories in these conditions were each modified
slightly to refer to the diagrams appropriately, just as in Experiment 3.
The procedure of Experiment 6 was essentially the same as that of Experiment 5. Subjects
first studied the stories for live min and then summarized them and described their similarities from memory. Subjects in the with-diagrams conditions were also asked to draw thediagrams. Following the story task the radiation problem was administered as usual. The
subjects, 189 in all, were run in small groups; approximately equal numbers of subjects
served in each of the four conditions.

Results and Discussion. Table 9A presents the percentage of subjects ineach condition who produced the convergence solution before the hint,
after it, and in total. Provision of a diagram resulted in a higher frequency
of initial solutions (57 vs 37%), G2(1) = 7.55, p < .Ol, as well as of totalsolutions (92 vs 79%), G2(1) = 6.66, p < .Ol. This total solution frequency
obtained when subjects received two analogs plus the diagrams, 92%, ishigher than that observed in any previous experiment with the radiation

ANALOGICAL

29

TRANSFERTABLE 9A. Percentage of Subjects Producing Convergence
Solution (Experiment 6)

Similar analogs with diagrams
Dissimilar analogs with diagrams
Similar analogs without diagrams
Dissimilar analogs without diagrams

Before
hint

After
hint

Total

N

61 (29)53 (27)40 (17)34 (16)33 (16)37 (19)40 (17)45 (21)

94 (45)90 (46)79 (34)79 (37)

48
51
43
47

B. Percentage of Subjects at Each Levelof Schema Quality
Schema quality

Similar analogs with diagrams
Dissimilar analogs with diagrams
Similar analogs without diagrams
Dissimilar analogs without diagrams

Good

Intermediate

Poor

N

44 (21)65 (33)26 (11)30 (14)29 (14)18 (9)
30 (13)36 (17)27 (13)18 (9)
44 (19)34 (16)

48
51
43
47

Note. Frequencies given in parentheses.

problem: nine times higher than the base rate for generation of the solution without any analog, it is clearly close to the maximum any manipulation could achieve.5 The facilitating effects of the diagrams were essentially the same for similar and dissimilar analogs; analog similarity had no
significant influence on solution frequencies.
Subjects’ descriptions of the similarities between the two stories wereagain scored for schema quality. A few subjects in the with-diagrams
conditions explicitly referred to the diagrams as representing the similar
aspects of the two situations. We made the assumption that such subjects
understood the underlying principle, and scored them as having goodschemas. (If this assumption were unduly generous, it should reduce therelationship between schema quality and transfer performance for the
with-diagrams conditions.) Table 9B presents the percentages of subjects
at the three levels of schema quality, for each of the four conditions.
Provision of the diagrams improved the quality of the schemas derived
from both similar and dissimilar analogs, G2(2) = 14.4, p < .OOl. Of
subjects who received diagrams, 55% wrote good schemas and only 22%
wrote poor ones; of those who did not receive diagrams, only 28% gave
good schemas while 39% gave poor ones. Thus, once again, as in Experi5 In an earlier experiment using one pair of dissimilar analogs (Gick, Note l), the additionof the diagrams yielded solution frequencies of 68% without a hint and 95% in total (N = 37).
The results of this study were replicated in all comparable respects by those of Experiment 6.

30

GICK AND HOLYOAK

ment 5, a manipulation that facilitated schema induction also increased
the degree of analogical transfer.
When no diagrams were given, analog similarity did not influenceschema quality, G2(2) < 1. However, when the diagrams were presented,
there was a tendency for dissimilar analogs to yield a higher frequencyof good schemas than was the case for similar analogs (65 vs 44%).
This trend was in fact significant, G2(1) = 4.41, p < .05; it provides
a hint that a mapping between relatively disparate analogs, when guided
by a device that highlights the underlying solution principle, may be more
likely to generate the optimal schema. However, as noted above, thisincrease in the proportion of good schemas did not translate into superiortransfer performance for subjects given dissimilar analogs. The data reported below will shed light on this apparent discrepancy.
The data in Table 10 convey the relationship between schema quality
and success in generating the analogous solution to the transfer problem.
Because this pattern did not differ significantly as a function of whether
diagrams were provided, the data have been collapsed across that variable. When the prior analogs were dissimilar to each other, the usual
positive relationship was found: the better the schema, the more likely theconvergence solution was to be produced, especially prior to the hint,
G2(4) = 21.5, p < .OOl. But somewhat surprisingly, the relationship was
far less strong when the prior analogs were similar, G2(4) = 5.16, p = .27.
Some degree of positive relationship was still apparent, which was significant when the most extreme cells were compared (the frequencies withwhich solutions were produced before the hint vs not at all, for subjects
with good vs poor schemas), G2(1) = 3.98, p < .05.
TABLE 10Percentage of Subjects Producing Convergence Solutions at Each Stage as a Functionof Schema Quality and Similarity of the Prior Analogs (Experiment 6)
Timing of convergence solution
Schema quality

Before
hint

After
hint

Not
produced

Similar analogs
Good
IntermediatePoor

63 (20)44 (12)44 (14)31 (10)45 (12)34 (11)

11 (3)
22 (7)

32
27
32

Dissimilar analogs
Good
IntermediatePoor

64 (30)39 (10)12 (3)

28 (13)50 (13)56 (14)8 (4)11 (3)
32 (8)

47
26
25

Note.

Frequencies given in parentheses.

6 (2)

N

ANALOGICAL

TRANSFER

31

In fact, the only substantial difference in Table 10 between the patterns
obtained for similar vs dissimilar analogs involved the relative frequencies
with which subjects who wrote poor schemas produced the solution without a hint (44% for similar analogs vs 12% for dissimilar analogs). There isa sense in which our measure of schema quality establishes only a lower
bound on what subjects have learned from the prior analogs; they may
write down less than they know. Those who receive similar analogs may
sometimes write down only similarities irrelevant to the convergence
principle, and hence be scored as having poor schemas; nevertheless,they may have actually abstracted the critical correspondences to somedegree. This is less likely in the case of subjects who receive dissimilar
analogs; because there are few superficial resemblances to describe, those
who write poor schemas have probably really missed the basic analogy.
Thus, while dissimilar stories, when accompanied by the diagrams,
yielded a higher frequency of good schemas, those subjects who wrote
poor schemas on the basis of dissimilar analogs were less successful attransfer than were their counterparts who wrote poor schemas on thebasis of similar analogs. This pattern is in accord with our hypothesis
regarding analog similarity: while dissimilar analogs have greater potential
to yield optimal schemas, they are also more likely to fail to produce any
useful schema. At a descriptive level, this tradeoff accounts for the absence of differences in overall transfer performance as a function of similarity of the prior analogs.
GENERALDISCUSSION

Summary and Implications

The experiments in Part I attempted to foster the abstraction of a problem schema from a single story analog by means of summarization instructions, or else either verbal or visual statements of the underlying
principle. We found no evidence that any of these devices yielded more
abstract representations of the story, nor did any consistently facilitate
analogical transfer. In contrast, the results obtained in Part II were dramatically more positive. Once two prior analogs were given, subjects
often derived an approximation to the convergence schema as the incidental product of describing the similarities of the analogs; furthermore,
the quality of the induced schema was highly predictive of subsequenttransfer performance. In addition, the same verbal statements and diagrams that had failed to influence transfer from a single analog proved
highly beneficial when paired with two.
These central results, as well as other more detailed findings (e.g., theextreme separation of noticing from application in the case of a diagrammatic analogy presented alone) support our analyses of analogical

32

GICK

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HOLYOAK

mapping, schema formation, and semantically driven retrieval. A mapping process cannot operate on only a single prior analog to derive aschema; consequently, the most direct mechanism for schema induction
is inapplicable. Two analogs, on the other hand, can be mapped together
to derive a more general schema; furthermore, any device that highlights
the causally relevant correspondences will facilitate abstraction of a more
optimal schema. To the extent that the schema implicit in the prioranalogs has been made explicit, analysis of a disparate transfer problem
may yield semantic retrieval cues that prompt recall of the prior information. Schema induction will thus increase the probability that an analogy
will subsequently be noticed; in addition, a problem schema will simplify
the process of mapping the prior information with the new problem inorder to generate an analogous solution.
It should be noted that in all of our experiments the critical prioranalogs were presented in a context in which their problem-oriented
character was incidental. Subjects were never explicitly encouraged touse the stories to learn about a novel kind of problem. In many situations,such as an instructional context, more directive guidance in the application of an analogy is often given. It is quite likely that more intentional
learning procedures could improve transfer performance in our paradigm.
In particular, explicit guidance might facilitate transfer from a single
analog. In the absence of such guidance, failure to derive a generalschema from a single instance may only reflect appropriate conservatism;
without either further examples or direct instruction, the person may haveno principled way to isolate the essential causal aspects of the situation.
Given the difficulty of schema abstraction from a single analog (at least
without the guidance of a teacher), one might ask how anyone could
spontaneously notice an analogy between one initial analog and a semantically remote transfer problem. After all, we obtained consistent estimates that about 20% of our subjects solved the radiation problem byusing the military story without receiving an explicit suggestion to do so.
In part, this minority may simply consist of subjects sensitive to demand
characteristics that suggest the two parts of the experiment might berelated. However, there are more basic reasons why at least a few people
may spontaneously represent the initial story in a way that would enable
subsequent transfer. Our subjects may have often brought relevant priorknowledge to the experiment. One subject in an earlier experiment (Gick& Holyoak, 1980) immediately mapped the story analog with another,previously known convergence problem (in which multiple weak lasers
were used to fuse the filament of a lightbulb without breaking the surrounding glass). While few subjects are likely to be reminded of such acomplete analogy to the story, many may bring some type of prior knowledge to bear on it. For example, college students are likely to alreadyANALOGICAL

TRANSFER

33

know the abstract concept of “force,” and a few may spontaneously
encode a military attack as an instance of the application of force. If such
subjects later see the radiation problem as one involving a search for amethod of applying force, this shared semantic component may trigger
recall of the prior story and initiation of a mapping process. More generally, if a current situation can be related to prior knowledge, so that its
causally relevant aspects are encoded at an abstract level, then the situation has the potential to be subsequently related to a new analog from aremote domain. Because experts tend to encode problems at a relatively
abstract level (Chi, Feltovich, & Glaser, 1981), it follows that expert
knowledge of one domain should tend to enable analogical transfer toanother. (See Holyoak, in press, for discussion of the relationship between analogical thinking and problem-solving expertise.)
Given our emphasis on schema induction as a basis for analogicaltransfer, it is worth considering what, if anything, specific analogs can
add to a general schema. It might seem that if a schema could be somehow
directly taught, examples would not be of any use at all. However, thepresent study provides evidence against this hypothesis. The provision ofa verbal statement of the convergence principle without any accompanying story (Experiment 2) could be viewed as an attempt to teach theschema directly. The total percentage of subjects producing the target
solution in this condition was not exceptionally high (66%), at least whencompared to the levels of transfer obtained in the later experiments withmultiple analogs. This might seem to contradict our prediction that reasoning from a schema will be more effective than reasoning from ananalog. However, the discrepancy is only apparent. Our prediction refers
to the benefit provided by a schema abstracted from specific analogs;
there is no reason to assume that the verbal statement used in Experiment
2 conveys the same information as could be induced from the story
analogs. Indeed, the verbal principle does not in fact allow as complete a
mapping with the radiation problem as do the stories. For example, theprinciple refers to “accomplishing some purpose,” whereas the corresponding core idea in the stories is the more specific relation of “overcoming.” In addition, the stories may allow a more detailed mapping thatgenerates information about how the abstract convergence solution could
be realized in a medical situation. For instance, the use of multiple hoses
in “Red Adair” could suggest the idea of introducing multiple rayemitting machines.
More generally, the more detailed information implicit in the representation of a specific analog may in part account for the intuitive viewthat examples are often critical for learning; for a teacher to simply “explain the principle” is not enough. To be put to use, a general schemamust always be instantiated as a set of specific concepts appropriate to a

34

GICK

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HOLYOAKparticular domain; an example, even from a different domain, maysuggest how the process of instantiation could proceed. Furthermore, incomplex domains much of the detailed knowledge shared by experts,
particularly procedural knowledge, is likely to be implicit and not easily
verbalized. A teacher may therefore have difficulty explicitly teachingsuch knowledge. However, by presenting the student with selected
examples, the knowledge may be conveyed implicitly.
Toward a Computational

Model

A great deal remains to be learned before it will be possible to formulate
a computationally adequate model of analogical thinking. Without such amodel, it is difficult to know how to optimize people’s use of analogies.
For example, Polya (1957) suggests searching for similar examples as aproblem-solving heuristic; a worthy goal without doubt, but how to findthe relevant examples and apply them? Perhaps analogical thinking can be
developed by appropriate training; however, this educational goal ishandicapped by our lack of knowledge regarding the essential processes
underlying the skill. The mapping process presents particularly formidable problems that must be solved in order to construct a process model ofanalogical thinking, and computational implementations have tended to
invoke algorithms that would unduly strain the capacity limits of humanworking memory. For example, Evan’s (1968) program to solve proportional analogies between geometrical figures finds the optimal level ofabstraction for representing the figures by computing all possible levels.
A related mapping problem arises from the fact that the number ofpotential pairings between two sets of concepts quickly becomes verylarge as the number of concepts in each set grows. Winston’s (1980)
program to compare story scenarios must either test all possible matches,or else prune away all except those between concepts that share somearbitrary feature (e.g., characters of the same gender). As Winston notes,
this algorithm will never discover a male Cinderella. Psychological process models must strive to capture the remarkable flexibility that people
exhibit in their use of analogy. The curtains have scarcely parted on thisparticular window to the mind.
ANALOGAPPENDIX I
TO THE CORD PROBLEM
The Birthday(EXPERIMENT

1)

Party

It was Jane’s sixth birthday and her mother wanted to make it a very
special day for her. So she organized a big surprise party and invited theneighborhood children without Jane knowing about it. The plan was thatthe mother who usually picked up Jane after school would be late in order

ANALOGICAL

35

TRANSFER

to allow the children time to arrive before Jane. The big day finally arrived. Everything was just about ready, and it was fifteen minutes before
the children were supposed to come. Jane’s mother was putting the final
touches on the decorations for the party room. She was covering the walls
and ceiling with balloons and party streamers made of crepe paper andribbons. Jane’s mother was finishing up a decoration pattern. Two final
pieces were left that were dangling from the wood paneling above. She
had originally planned to knot these two pieces of ribbon together in order
to attach balloons to them. However, whenever she grabbed the end of
one ribbon, colored blue, she was not able to grasp the other ribbon,
colored pink, at the same time. The ribbons simply were not long enoughto be knotted together in this way. It seemed that she would have toabandon her final bit of decoration.
Jane’s mother was just about to give up when she had an idea. She took
the pair of scissors that she had been using to cut the various ribbons andcrepe paper, and attached the scissors to the end of the ribbon. Next, shetook hold of the scissors, and pointing them in the direction of the pinkribbon, swung them vigorously so that this blue ribbon now swayed alternately between the pink ribbon and a nearby wall. She then ran quickly
and took the end of the dangling pink ribbon and walked as close to theswinging blue ribbon as possible without letting go of the pink ribbon. She
then waited until the swinging blue ribbon came her way and caught it on
the upswing. While still holding the pink ribbon, she then removed thescissors from the other blue ribbon and knotted the two ribbons together.
Jane’s mother just managed to attach all her balloons on these ribbons,completing her decorations, before the children started pouring in. Soon
Jane arrived and was genuinely surprised. The party was a great success,and all the mothers complemented Jane’s mother on the decorations.
FOUR ANALOGSAPPENDIX II
TO THE RADIATIONPROBLEM

The General

A small country was ruled from a strong fortress by a dictator. The
fortress was situated in the middle of the country, surrounded by farms
and villages. Many roads led to the fortress through the countryside. A
rebel general vowed to capture the fortress. The general knew that an
attack by his entire army would capture the fortress. He gathered hisarmy at the head of one of the roads, ready to launch a full-scale direct
attack. However, the general then learned that the dictator had planted
mines on each of the roads. The mines were set so that small bodies ofmen could pass over them safely, since the dictator needed to move his
troops and workers to and from the fortress. However, any large force

36

GICK

AND

HOLYOAKwould detonate the mines. Not only would this blow up the road, but itwould also destroy many neighboring villages. It therefore seemed impossible to capture the fortress.
However, the general devised a simple plan. He divided his army into
small groups and dispatched each group to the head of a different road.
When all was ready he gave the signal and each group marched down a
different road. Each group continued down its road to the fortress so thatthe entire army arrived together at the fortress at the same time. In thisway, the general captured the fortress and overthrew the dictator.
The Commander

A military government was established after the elected government
was toppled in a coup. The military imposed martial law and abolished all
civil liberties. A tank carp commander and his forces remained loyal tothe overthrown civilian government. They hid in a forest waiting for achance to launch a counterattack. The commander felt he could succeed if
only the military headquarters could be captured. The headquarters was
located on a heavily guarded island situated in the center of a lake. The
only way to reach the island was by way of several pontoon bridges thatconnected it to the surrounding area. However, each bridge was so narrow and unstable that only a few tanks could cross at once. Such a small
force would easily be repulsed by the defending troops. The headquarters
therefore appeared invincible.
However, the tank commander tried an unexpected tactic. He secretly
sent a number of tanks to locations near each bridge leading to the island.
Then under cover of darkness the attack was launched simultaneously
across each bridge. All of the groups of tanks arrived on the island together and immediately converged on the military headquarters. They
managed to capture the headquarters and eventually restore the civilian
government.
Red Adair

An oil well in Saudi Arabia exploded and caught fire. The result was a
blazing inferno that consumed an enormous quantity of oil each day. Afterinitial efforts to extinguish it failed, famed firefighter Red Adair was called
in. Red knew that the fire could be put out if a huge amount of fire
retardant foam could be dumped on the base of the well. There wasenough foam available at the site to do the job. However, there was no
hose large enough to put all the foam on the fire fast enough. The small
hoses that were available could not shoot the foam quickly enough to do
any good. It looked like there would have to be a costly delay before aserious attempt could be made.
However, Red Adair knew just what to do. He stationed men in a circle

ANALOGICAL

TRANSFER

37

all around the fire, with all of the available small hoses. When everyone
was ready all of the hoses were opened up and foam was directed at thefire from all directions. In this way a huge amount of foam quickly struck
the source of the fire. The blaze was extinguished, and the Saudis weresatisfied that Red had earned his three million dollar fee.
The Fire Chief

One night a fire broke out in a wood shed full of timber on Mr.
Johnson’s place. As soon as he saw flames he sounded the alarm, andwithin minutes dozens of neighbors were on the scene armed with buckets. The shed was already burning fiercely, and everyone was afraid that if
it wasn’t controlled quickly the house would go up next. Fortunately, theshed was right beside a lake, so there was plenty of water available. If alarge volume of water could hit the fire at the same time, it would be
extinguished. But with only small buckets to work with, it was hard to
make any headway. The fire seemed to evaporate each bucket of water
before it hit the wood. It looked like the house was doomed.
Just then the fire chief arrived. He immediately took charge and organized everyone. He had everyone fill their bucket and then wait in a circle
surrounding the burning shed. As soon as the last man was prepared, thechief gave a shout and everyone threw their bucket of water at the tire.
The force of all the water together dampened the fire right down, and it
was quickly brought under control, Mr. Johnson was relieved that his
house was saved, and the village council voted the tire chief a raise in pay.
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NOTES

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